Dynamic Programming of Stochastic Burgers Equation Driven by Levy Noise

TL;DR

This paper develops a dynamic programming framework for optimal control of a stochastic Burgers equation driven by Levy noise, solving the associated Hamilton-Jacobi-Bellman equation with integro-differential operators.

Contribution

It introduces a novel approach to control stochastic PDEs with Levy noise using second-order HJB equations and regularization techniques.

Findings

Successfully solves the HJB equation for the stochastic Burgers equation with Levy noise.

Establishes a method for deriving feedback controls in stochastic PDEs with jumps.

Demonstrates the regularizing effects of the transition semigroup on the control problem.

Abstract

In this work, we study the optimal control of stochastic Burgers equation perturbed by Gaussian and Levy type noises with distributed control process acting on the state equation. We use the dynamic programming approach for the second order Hamilton-Jacobi- Bellman (HJB) equation consisting of an integro-differential operator with Levy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic Burgers equation and compactness arguments, we solve the HJB equation and the resultant feedback control problem.